Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

(26.5) cos^(2)(2x)-2cos(2x)=-1. Solve on [0,2pi]

(2626.55) cos⁡2(2x)−2cos⁡(2x)=−1 \cos ^{2}(2 x)-2 \cos (2 x)=-1 . Solve on [0,2π] [0,2 \pi]

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Find derivatives of sine and cosine functionsright chevron icon

Full solution

Q. (2626.55) cos⁡2(2x)−2cos⁡(2x)=−1 \cos ^{2}(2 x)-2 \cos (2 x)=-1 . Solve on [0,2π] [0,2 \pi]
  1. Recognize as quadratic equation: Recognize the equation as a quadratic in terms of cos⁡(2x)\cos(2x). Let y=cos⁡(2x)y = \cos(2x). Then the equation becomes y2−2y=−1y^2 - 2y = -1.
  2. Rearrange to standard form: Rearrange the equation to standard quadratic form.\newliney2−2y+1=0y^2 - 2y + 1 = 0
  3. Factor the quadratic: Factor the quadratic equation.\newline(y−1)2=0(y - 1)^2 = 0
  4. Solve for y: Solve for y.\newliney−1=0y - 1 = 0\newliney=1y = 1
  5. Substitute back for yy: Substitute back cos⁡(2x)\cos(2x) for yy.cos⁡(2x)=1\cos(2x) = 1
  6. Solve for 2x2x: Solve for 2x2x within the interval [0,4π][0, 4\pi], since 2x2x has twice the period of xx.\newline2x=0+2kπ2x = 0 + 2k\pi or 2x=π+2kπ2x = \pi + 2k\pi, where kk is an integer.
  7. Solve for x: Solve for x by dividing by 22.\newlinex=02+kπx = \frac{0}{2} + k\pi or x=π2+kπx = \frac{\pi}{2} + k\pi\newlinex=kπx = k\pi or x=π2+kπx = \frac{\pi}{2} + k\pi
  8. Find solutions within interval: Find all solutions within the interval [0,2π][0, 2\pi].\newlineFor x=kπx = k\pi, the solutions are x=0x = 0, π\pi, 2π2\pi.\newlineFor x=π2+kπx = \frac{\pi}{2} + k\pi, the only solution within [0,2π][0, 2\pi] is x=π2x = \frac{\pi}{2}.
  9. Combine all solutions: Combine all the solutions.\newlineThe solutions are x=0x = 0, π2\frac{\pi}{2}, π\pi, 2π2\pi.

More problems from Find derivatives of sine and cosine functions

Question
Find the derivative of f(x) f(x) .\newlinef(x)=cos⁡(x) f(x) = \cos(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡(x) f(x) = \tan(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ex f(x) = e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ln⁡(x) f(x) = \ln(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡−1x f(x) = \tan^{-1}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=xex f(x) = x e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=exx f(x) = \frac{e^x}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) . \newline f(x)=e(x+1) f(x) = e^{(x + 1)} \newline f′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=x+3 f(x) = \sqrt{x+3} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of g(x) g(x). \newline g(x)=ln⁡(2x) g(x) = \ln(2x) \newline g′(x)= g^{\prime}(x) = ______
Get tutor helpright-arrow

Posted 10 months ago

View all (129)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant